Multimodal Model Evaluator

Find the least odd prime factor of 201 9 8 + 1 2019^8+1 201 9 8 + 1 . To find the least odd prime factor of (2019^8 + 1), we start by noting that if a prime ( p ) divides (2019^8 + 1), then (2019^8 \equiv -1 \pmod{p}). This implies (2019^{16} \equiv 1 \pmod{p}), so the order of 2019 modulo ( p ) is 16. By Fermat's Little Theorem, the order must divide ( p - 1 ), hence ( p \equiv 1 \pmod{16} ). The primes congruent to 1 modulo 16 are 17, 97, 113, 193, etc. We check the smallest primes: Prime 17 : Compute (2019 \mod 17): (2019 \equiv 13 \pmod{17}) Calculate (13^8 \mod 17): (13^2 \equiv 16 \pmod{17}) (13^4 \equiv 16^2 \equiv 1 \pmod{17}) (13^8 \equiv 1^2 \equiv 1 \pmod{17}) Thus, (2019^8 + 1 \equiv 1 + 1 = 2 \pmod{17}), so 17 is not a factor.